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June 5, 2009

The Elusive Proteus

Posted by David Corfield

Cassirer writes in Determinism and Indeterminism in Modern Physics:

At first glance the universality of the action principle seems by no means beyond question. This universality could be attained only at the cost of a circumstance which, from the purely physical point of view, led again and again to difficulties and doubts. For the more universally the principle was conceived, the more difficult it became to specify clearly its proper concrete content. It becomes finally a kind of Proteus, displaying a new aspect on each new level of scientific knowledge. If we ask what precisely that “something” might be to which the property of a maximum or a minimum is ascribed, we receive no definite and unambiguous answer. (p. 50)

…can foretell the future, but, in a mytheme familiar from several cultures, will change his shape to avoid having to; he will answer only to someone who is capable of capturing him. From this feature of Proteus comes the adjective protean, with the general meaning of “versatile”, “mutable”, “capable of assuming many forms”: “Protean” has positive connotations of flexibility, versatility and adaptability.

Striking then how Saunders Mac Lane describes mathematics as ‘Protean’ in this paper.

The thesis of this paper is that mathematics is protean. This means that one and the same mathematical structure has many different empirical realizations. Thus, mathematics provides common overarching forms, each of which can and does serve to describe different aspects of the external world. (p. 3)

On the face of it, however, there’s a difference between Cassirer’s multiple-realisability and Mac Lane’s. The latter appears to be talking about a common structure reappearing with little disguise from one place to the next in the empirical world. But the examples Mac lane gives suggest he’s also thinking about the unpredictable unfolding of an idea within mathematics itself.

For the Galoisian idea he begins with Lagrange in 1770 finding resolvants for equations, and ends with ‘Various’ in 1990

One adjunction handles Galois and much more.

It is perhaps odd that he doesn’t include Poincaré and his deck transformations in this line of thought, although he does have Grothendieck in 1961 introducing Galois theory for covering spaces. And he might have said ‘covering spaces for number theory’.

So how’s that for a Protean principle in mathematics to rival the action principle in physics? Whatever that something is which has manifested itself most recently in nn-fibrations and fundamental groups in number theory, or can it be given a ‘definite and unambiguous form’?

Posted at June 5, 2009 1:56 PM UTC

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Re: The Elusive Proteus

‘…the adjective protean, with the general meaning of “versatile”, “mutable”, “capable of assuming many forms”’

A complete aside but… Is there an etymological reason that protons are named as they are? The ancient alchemists considered Proteus as a symbol of First Matter. (There are references to that in Milton’s ‘Paradise Lost’.)

Re: The Elusive Proteus

Yes, the people who named the proton were guessing that the proton was the fundamental constituent of all atoms. (Because of the fact that most elements seemed to have atomic weights that were very close to integer multiples of that of hydrogen, people had been wondering aloud since the beginning of the 19th century whether all elements were fundamentally made of hydrogen in some way, and this was a continuation of that thought.)

Re: The Elusive Proteus

I think the word ‘proton’ was based on the Greek work protos, meaning ‘first’, rather than the name Proteus. Protons aren’t protean, they’re just fundamental.

But apparently Proteus may mean something like ‘primordial’ or ‘firstborn’, so there’s a relation.

…I’d rather be
A Pagan suckled in a creed outworn;
So might I, standing on this pleasant lea,
Have glimpses that would make me less forlorn;
Have sight of Proteus rising from the sea.
Or hear old Triton blow his wreathèd horn.

Re: The Elusive Proteus

I think the word “proton” was based on the Greek work protos

Oops, yes, sorry, that’s what I meant. There is an “etymological reason why protons are named as they are”, and it does have, indirectly, to do with the idea of First Matter, but the word “Proteus” is a bit of a side-alley in the matter. Sorry to be so hopelessly unclear.

(Actually, wouldn’t the word “proton” be based on the Greek word “proton” (πρωτℴν\pi\rho\omega\tau\omicron\nu), which would be (Greek speakers correct me if I am wrong) the neuter form for which “protos” is the masculine?)

Re: The Elusive Proteus

Tim wrote:

Actually, wouldn’t the word “proton” be based on the Greek word “proton” (πρωτℴν), which would be…

Quite possibly right. You’ve out-erudited me.

Of course I didn’t really think you thought protons were named after Proteus, but I was afraid Johan Alm might have picked up this impression from what you said, given that we were talking about Proteus.

Re: The Elusive Proteus

Constituting Objectivity: Transcendental Perspectives on Modern Physics (The Western Ontario Series in Philosophy of Science) (Hardcover)
by Michael Bitbol (Editor), Pierre Kerszberg (Editor), Jean Petitot (Editor)

This book costs $259, however I did learn the difference between transcendent and transcendental :-)

“The mere fact that physics involves a computational synthesis of observable
phenomena means that physical objectivity cannot be tantamount to an ontology
of some independent substantial reality. Indeed, the possibility of a mathematical
reconstruction of such an ontological reality would ascribe the human mind
excessive intellectual capacities which transcend its finiteness. This leaves only
two options:
(a) Physics is purely descriptive. It conceptually organizes the empirical manifold
by means of an Analytic, and it can thereby pretend it describes an ontological
independent reality, but without reconstructing this reality mathematically and
without doing any job other than picturing it passively (empiricism +
nominalism).
(b) Physics can reconstruct the empirical manifold mathematically, and it must then
accept to partake of a “weak” form of objectivity which de jure can only con-
cern relations between observable phenomena, namely a reality filtered by ineliminable conditions of experimental or sensory accessibility, and by intellectual
criteria of selection. The condition of possibility of computational synthesis is the
principle of restriction of physical knowledge to laws of observables, and the
decoupling between a “strong” ontology and a “weak” objectivity.”

Re: The Elusive Proteus

I made an attempt to pin down that elusive Proteus — the basic idea of Galois theory — in a talk I gave a few years back. But Jim Dolan thinks I blurred the precise relationship between Galois theory and the ‘Erlangen program’ idea, and he’s probably right. They’re really the same thing, but only if you think of it the right way. And I didn’t even mention the Beth definability theorem, which is yet another guise of this Proteus. So, think of it as a rough draft.

Re: The Elusive Proteus

Perhaps I’ve never quite understood the Erlangen program, but I wonder if you could explain a bit why these are the same thing. Among arithmetic π1\pi_1-theorists there has been some discussion about this connection, but I think we never were able to see in some precise sense that the two had a uniform formulation. If you have some insight on this issue, I’d quite like to hear of it!

Re: The Elusive Proteus

The emphasis in that lecture is on the correspondence between subthings and isotropy groups. But I thought the deeper part of the Erlangen programme had to do with how the *geometry itself* can be recovered from its automorphism group, which of course doesn’t work when we consider geometry in the 20C sense.

In any case, the comparison I had in mind was with this part of the programme (as I understood it, anyways) and Grothendieck’s anabelian programme, whereby certain schemes can be recovered from their fundamental groups. For example, given two number fields FF and KK, if

Gal(F¯/F)≃Gal(K¯/K),Gal(\bar{F}/F)
\simeq Gal(\bar{K}/K),

then

F≃K.F\simeq K.

Furthermore, the isomorphism of groups is induced (up to inner automorphism) by an isomorphism of fields. Another way of saying this is that FF can be recovered from its category of covering spaces.

I think we discussed these matters at some point, but would you happen now to have any thoughts about such statements?

NB: Here is one striking instance of the anabelian theorem about number fields:

Re: The Elusive Proteus

Minhyong wrote:

But I thought the deeper part of the Erlangen programme had to do with how the *geometry itself* can be recovered from its automorphism group…

As far as I can tell, in the Erlangen program the ‘geometry itself’ is roughly just the category of all GG-sets, where GG is the automorphism group of the geometry. Or even better, it’s the bicategory of GG-sets, spans of GG-sets, and maps of spans. (Of course this bicategory can be constructed from the category of GG-sets.)

Jim and I call this bicategory the Hecke bicategory.
This bicategory nicely contains all the information about ‘incidence relations’ between ‘types of figures’.

…which of course doesn’t work when we consider geometry in the 20C sense.

Well, geometry is an elusive Proteus if there ever was one! Until it gets a booster shot, the Erlangen program is only about a special sort of geometry: ‘incidence geometry’. This should be general enough to have lots of deep relations to Galois theory. But I have no clue as to why a number field should be recoverable from its Galois group. This amounts to saying there’s nothing more to it than its ‘incidence geometry’.

Re: The Elusive Proteus

In geometry, you start with a space, then find the automorphism group, GG. Then you realise that quotienting that group by the stabilizer of a point gives you your original space. So you then figure out that the same trick should work for stabilizers of other figures. This leads you to recognise that you could have started out simply with GG. The category of GG-sets defines a geometry with incidence relations encoded. For some groups GG there’s an old-fashioned interpretation in terms of points, lines, etc.

John tells us above about his Hecke bicategory construction. Two questions: Does that construction work for Lie groups? Is there a Hecke tricategory relating the Hecke bicategories for different GG?

Now what happens in the Galoisian case? Given a group, HH, of automorphisms of a field, FF, we don’t just go hunting for any old HH-sets. At the undergraduate level at least, we want HH to act on field extensions of the ground field. So is there a way of forgetting my original FF, and just looking at some category of HH actions on some kind of structures.

In the latter case the things acted on have at least a vector space structure. John wrote a fictional discussion between Frobenius and Burnside in episode 5 of groupoidification aimed at showing us how his Hecke bicategory compares with ordinary group representations into Vect. What’s the final score there? Does your bicategory have all the information of the vector space representations?

Re: The Elusive Proteus

I’m not sure I understand the question, but if H=Gal(F/K)H=Gal(F/K), for some finite Galois extension FF of KK, then there will indeed be a correspondence between *arbitrary* pointed finite transitive HH-sets and intermediate fields. This is the usual thing whereby to any intermediate field

K⊂L⊂FK\subset L\subset F

we associate the transitive HH-set of KK-embeddings L↪FL\hookrightarrow F with the original embedding as base point, whereas for a transitive pointed HH-set SS, the stablizer of the point will provide an intermediate fixed field. If we leave out the point, we get the field extensions of KK that *admit* an embedding into FF.

When KK happens to be a number field and FF is not a finite extension but the algebraic closure of KK, then KK itself can be recovered from H=Gal(F/K)H=Gal(F/K) and the category of direct sums of finite extensions from the finite sets with continuous HH-action.

Re: The Elusive Proteus

I’m not sure I understand the question…

Maybe I don’t either, but is there not a sense in which there isn’t such a radical ‘letting go’ in the number theoretic case?

In the geometric case you really can forget that you ever got to consider a group just because it happened to be the group of automorphisms of a space you had already been thinking of. Instead, you can hand me an arbitrary Lie group, and I can say it determines a ‘geometry’. If it’s a certain kind of group (with parabolic subgroups in the right way), I may then be able to tell you a story of point on lines, lines on planes, etc.

How can we generalize this idea to arbitrary complex simple Lie groups? The trick is to follow Felix Klein’s “Erlangen program” relating geometry and group theory. There are many kinds of geometry, each with its own symmetry group. In a geometry with symmetry group GG, different types of figure correspond to different subgroups of GG. The idea is that for each type of figure, there is a space XX of all figures of that type, upon which GG acts. Given any two of these figures, there’s some element of GG mapping one to the other: that’s what we mean by saying they’re of the same type! You can show this implies there’s an isomorphism

X=G/H
X = G/H

for some subgroup HH of GG. HH is the group of transformations that preserves a given figure – that is, maps it to itself.

Conversely, any subgroup HH can be thought of as determining a type of figure! But in practice, some subgroups correspond to more familiar types of figure than others. In particular, every complex simple Lie group GG has certain “maximal parabolic subgroups” coming from the dots in its Dynkin diagram of GG, and these give the types of figure that we really want to understand.

In number theory do you ever free yourself from the knowledge that ‘really’ you’re dealing with specific number fields and their extensions? Could you?

Say I’m thinking of ℚ(2,i)\mathbb{Q}(\sqrt{2}, i). You calculate the Galois group, GG, as the Klein four group. Now could you forget where it came from and look for representations in the category of fields?

It feels to me more like the case with groups and representations in vector spaces. There you don’t want transitive actions at the level of elements of the vector space, but rather linear actions. But there you care about products of representations and how they decompose into irreducibles, something you couldn’t do with fields.

Re: The Elusive Proteus

Now I understand what you were asking about and I agree entirely. The framework of classical Galois theory is indeed very weak when it comes to encoding a field into a group in whatever sense. This was the point of my initial comment in this thread, whereby Galois theory couldn’t really be compared to the Erlangen programme in the sense of ‘recovering a geometry from its automorphisms.’ As you note, at best we have a field FF and a group GG of automorphisms with corresponding fixed field KK. Then intermediate fields correspond to subgroups. The reason I’m recalling this elementary statement is to point out one further inadequacy: There might be two subgroups H1,H2H_1, H_2 which are isomorphic as groups, but for which FH1F^{H_1} and FH1F^{H_1} are not isomorphic. That is, even if we are given an ambient field FF to start with, it is essential to have the data of subgroups, not ‘groups that are isomorphic to subgroups.’

If we consider in turn the point of view of KK, then it corresponds to the group GG in this picture, but of course this correspondence will depend on the choice of an FF that contains it.

“In number theory do you ever free yourself from the knowledge that ‘really’ you’re dealing with specific number fields and their extensions? Could you?”

It is this question that is addressed by anabelian geometry. If we go a different route and look instead at the category of *all* coverings of a number field KK, then it does characterize KK (among number fields). This is the sense in which anabelian geometry overcomes some deficiencies of classical Galois theory, and can be compared with EP. From the view of groups, K¯\bar{K} is an intrinsic invariant of KK, and the group GK=Gal(K¯/K)G_K=Gal(\bar{K}/K) characterizes KK. Thus, the problem of group-theoretically capturing a space KK with very few automorphisms (whence the degeneracy you mention), even none at all, has been resolved. Incidentally, one can move beyond number fields to all finitely-generated fields. In this picture, the separably closed fields of finite transcendence degree will correspond to various homogeneous spaces, while finitely-generated fields are like locally homogeneous spaces.

An open problem here is to understand what class of GKG_K can arise this way. At present, we have no idea, so the situation is unsatisfactory compared to our understanding of finite volume discrete subgroups of Lie groups. My friend Shinichi Mochizuki regards it as immensely important to figure out how to ‘construct directly’ KK from GKG_K. Of course the vagueness of the problem is acknowleged by the quotes. Very roughly speaking, it should be reminiscent of how the operations in a field can be reconstructed from the geometry of a projective space over that field. A resolution of this problem should be intimately related to the problem of characterizing the GKG_K. Even more speculatively, perhaps it will suggest *category-theoretic generalizations* of number fields.

Re: The Elusive Proteus

David wrote approximately:

John wrote a fictional discussion between Frobenius and Burnside in episode 5 of The Tale of Groupoidification aimed at showing us how his Hecke bicategory compares with ordinary group representations into Vect. What’s the final score there? Does your bicategory have all the information of the vector space representations?

The quick answer is ‘yes’.

The pompously named Fundamental Theorem of Hecke Algebras gets us part of the way. Starting with a finite group GG we get the Hecke bicategory Hecke(G)Hecke(G), with

finite GG-sets as objects,

spans of finite GG-sets as morphisms,

maps of spans as 2-morphisms.

In HDA8 we’ll explain how to wave a magic wand over Hecke(G)Hecke(G) and get Perm(G)Perm(G), the category of permutation representations of GG. These are linear representations that arise from actions of GG on finite GG-sets, so it’s not surprising.

In general, not every finite-dimensional rep of GG is a permutation representation, so

Perm(G)≇Rep(G)Perm(G) \ncong Rep(G)

However, at least over the complex numbers, we can get Rep(G)Rep(G) from Perm(G)Perm(G) by waving another magic wand over it: ‘splitting idempotents’. This is a general category-theoretic construction. The rough idea is that given a morphism p:V→Vp: V \to V with p2=pp^2 = p, we make up (if necessary) a new object Ran(p)Ran(p) that acts like the ‘range’ of pp.

Since we can get Rep(G)Rep(G) from Hecke(G)Hecke(G) by waving two magic wands over it, there’s a sense in which the Hecke bicategory has all the information about linear representations of GG tucked away inside it.

But more important, as far as I’m concerned, is how we can beautifully lift a number of interesting constructions from the category Rep(G)Rep(G) to the bicategory Hecke(G)Hecke(G). So, to some limited extent, we can argue that a bunch of stuff people do with group representations is ‘really’ stuff about the Hecke bicategory. I wouldn’t want to go overboard on this claim, but it’s a fruitful attitude.

Re: The Elusive Proteus

Looking at spans of linear representations of a group not my kettle of tea, since my goal in working with spans of GG-sets is to reveal the linear representation theory of finite groups as a shadow of something deeper and simpler.

A span of sets or groupoids mimics a matrix of numbers. My dream is to treat spans of sets and groupoids as ‘more primordial’ than matrices of numbers. So, I’ve little interest in pondering spans in the world of linear algebra.

The idea of a span of sets is that, unlike a map of sets but like a matrix, we can turn it around, or ‘transpose’ it. We can also add spans of sets.
It’s these features that make spans of sets act like linear maps between finite-dimensional Hilbert spaces.

Note that spans in the world of finite-dimensional Hilbert spaces can be turned into linear maps, since we can turn around one arrow and then compose the two arrows. So, at least in some vague sense, spans in the world of finite-dimensional Hilbert spaces are ‘redundant’. Maybe there’s an equivalence of (bi)categories that would make this idea precise.

Re: The Elusive Proteus

I was just wondering if we could say that taking spans is the generic thing to do. Because VectVect has some special feature it turns out that we don’t need to do the span construction for representations of a group GG there, and can stick with Rep(G)Rep(G). SetSet lacks that special feature so the collapse to G−SetG-Set loses information.

To really get to know GG wouldn’t we want to see how it can be representated in all categories? And generically we would need the Hecke bicategory for each category. Or are you hoping SetSet will do for all GG’s needs?

Re: The Elusive Proteus

Sure!

Sticking to finite groups is just a way to illustrate certain ideas in a simple way. Right now Alex Hoffnung is working on ‘change of base for bicategories enriched over a monoidal bicategory’, for the purposes of writing HDA8. It’s quite amusing how such ideas are downright natural and necessary for a modern understanding of finite groups and their actions!

But anyway, here we have a lot of algebra but no ‘smoothness’ to worry about. If we brought in smoothness and categorified further, we’d pushed towards things that the geometric ∞\infty-function theory crowd are interested in. Which is great stuff, but there’s a certain charm to keeping things simple.

Re: The Elusive Proteus

Generalizing the Erlangen programme to more general geometries seems to me a good project for category theorists and perhaps you’ve already made a good deal of progress. But let me quickly summarize my thoughts anyways.

Presumably the set up we are considering provisionally is where we start with some kind of geometry XX, and associate to it an `algebraic structure’ G(X)G(X) that characterizes XX in some suitable sense. G(X)G(X) might be the automorphisms. These days of course, we think of this particular gadget as working only for *geometries with many automorphisms*, that is, homogeneous ones. I’m very unsystematic in my understanding of even this but I do know that this works for two and three manifolds, the latter case being contained in Thurston’s classification. The question then is what to do when there are very few automorphism or none at all, as is the case for the generic compact two manifold. As an intermediate step to a general question, one could consider, for example, locally homogeneous spaces, i.e., spaces whose universal covers are homogeneous, which includes all compact two manifolds. They form also the building blocks of compact three manifolds according to Thurston’s programme. In this case, it seems that G(X)G(X) should be a suitable category of principal bundles for the fundamental group. The reason is that if we pick a basepoint bb, then the union of all principal bundles of paths emanating from bb *is* the universal covering space, which can then be `recovered’ from its group of automorphisms. The original space can then be recovered from this union and the trivial principal bundle acting on it. So when XX is a locally homogeneous space, we should have

(1) π:=π1(X,b)\pi:=\pi_1(X,b);

(2) A category of principal bundles for π\pi that is rigid enough to give us exactly the bundles π1(X;b,x)\pi_1(X;b,x) for the points x∈Xx\in X;

(3) Some way of thinking of this category that allows us to recover the metrized automorphisms of

X˜=∪xπ1(X;b,x).\tilde{X}=\cup_x \pi_1(X;b,x).

together with π\pi as a subgroup.

This will then give us XX back. A suitable way of carrying this out should be closely related to the anabelian programme as well as my own interest in the connection to Diophantine equations.

Re: The Elusive Proteus

The question then is what to do when there are very few automorphisms or none at all…

If the geometric space is non-homogeneous, shouldn’t symmetries of spaces be captured better by groupoids, as Weinstein explains?

The classic Erlanger situation is a Lie group treated as a one-object category mapping into the category of manifolds. The action is transitive. The action groupoid is connected. The vertex group for any object is a closed Lie subgroup.

Generalizing, why not start with a Lie groupoid and map it into Manifolds?

Re: The Elusive Proteus

Last Action Extraterrestrials; Re: The Elusive Proteus

The Heptapods in Ted Chiang’s “Story of Your Life,” which depends on an ultra-strong version of the Sapir-Whorf hypothesis, and a non-standard viewpoint on Time in a Block Universe, puts the least-action principle at the center of their science (arguably the most interesting idea of the story).